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1. Consider the following simple linear regression modelYi =?0+?1Xi+?i

with ?i independent normall-distributed errors with E [?i] = 0 and Var[?i] =?2, for i = 1,…,n (e.g. ?i ? N(0,?2)). A common use of the regression model is to estimate the value of the outcome Y conditional upon a given valueofX. AnaturalestimateofE[Y|X =Xh]isgivenbyYˆh:

Yˆh = ?ˆ0+?ˆ1Xh
= Y? ? ?ˆ 1 X? + ?ˆ 1 X h= Y?+?ˆ1(Xh?X?)

(a)  FindtheexpectationofYˆh.IsYˆhanunbiasedestimatorofE[Y|X=Xh]? (b)  Show that the variance of Yˆh is given by

n ? ni = 1 ( X i ? X? ) 2(Hint:youwillneedtousethefactthatCovY?,?ˆ =0.

(c) WhatisthedistributionofYˆhundertheassumptionsofthemodelgiven above?

(d) Suppose we estimate ?2 by s2 = RSS , derive the distribution ofn?2

1 (X?X?)2 ?2 + h

Yˆ h ? E [ Y | X = X h ]1 (Xh?X?)2

1

s2 n + n ? 2?i=1 (Xi ?X )

(Hint: you can use the fact that RSS ? ?2 that we mentioned in class

?2 n?2
(e) Usingyourresultffrom(d),howyouestimatea95%CIforE[Y|X=Xh]?

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